Q.$\int \dfrac{x^4 - 1}{x^2 + 1}\,dx =$

Polynomial Long Division

Dividing $137$ by $4$ asks "how many 4's fit into 137?" — answer $34$, remainder $1$. Polynomial long division is the same question with variables: how many times does the divisor fit into the dividend? You get a quotient polynomial plus a remainder whose degree is smaller than the divisor's. The only change from arithmetic is that you compare the highest power of the variable instead of place value.

This is the Division Algorithm for Polynomials.

The routine

To divide $P(x)$ by $D(x)$, repeat until the remainder's degree drops below $\deg D$:

1. Divide the leading term of the current dividend by the leading term of $D(x)$ — this is the next quotient term.
2. Multiply the whole divisor by that term.
3. Subtract to get a new, lower-degree dividend, then repeat.

For example, dividing $2x^3 + 3x^2 - 5x + 1$ by $x-2$: the successive quotient terms are $2x^2$, then $7x$, then $9$, leaving remainder $19$. So

The remainder $19$ has degree $0 < 1$, exactly as the algorithm requires.

Why it matters

• If $R(x)=0$, then $D(x)$ is a factor of $P(x)$. …